Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the field of vehicle design, finance, etc. Here, we will discuss the square root of -56.
The square root is the inverse of the square of the number. Since -56 is a negative number, its square root involves imaginary numbers. The square root of -56 is expressed in both radical and exponential form. In the radical form, it is expressed as √(-56), whereas (-56)^(1/2) in exponential form. The principal square root of -56 is expressed as √(-56) = 7.48331i, where i is the imaginary unit, because the square root of a negative number is not a real number.
For negative numbers, the square root involves imaginary numbers. The methods used for finding square roots of positive numbers are not directly applicable. However, we can express the square root of a negative number using the imaginary unit i. The steps are as follows:
1. Separate the negative sign from the number.
2. Find the square root of the positive part.
3. Multiply the result by i, the imaginary unit.
Prime factorization is typically used for non-negative numbers. However, we can adopt the process to find the square root of the positive part:
Step 1: Finding the prime factors of 56 Breaking it down, we get 2 x 2 x 2 x 7: 2^3 x 7^1
Step 2: Now, we found the prime factors of 56. Since -56 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely.
The square root of -56 is expressed as √(-56) = √(56) x i = √(2^3 x 7) x i = 2√(14) x i.
The long division method is typically used for positive numbers, so we adapt it for the positive part of -56:
Step 1: Consider the positive part, 56, and group the numbers from right to left.
Step 2: Find n whose square is closest to 56. For 56, n is approximately 7, since 7^2 = 49.
Step 3: Calculate further for better approximation if necessary. However, since we aim for the imaginary square root, the interest is more in identifying the pattern of the imaginary unit: √(-56) = √(56) x i = 7.48331i.
The approximation method provides a quick way to find the square root of the positive part before applying the imaginary unit:
Step 1: Identify √56, which lies between √49 (7) and √64 (8).
Step 2: Use the approximation formula (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). For √56: (56 - 49) / (64 - 49) = 7/15 = 0.4667 Add this to 7: 7 + 0.4667 = 7.4667
So, √(-56) = 7.4667i
Students often make mistakes when dealing with square roots of negative numbers, especially by ignoring the imaginary unit or incorrectly applying real number methods. Below are common mistakes and how to avoid them.
If a square has an area of -56 square units, what is the side length?
The side length is 7.48331i units.
The area of a square is side^2.
For an area of -56, side = √(-56) = 7.48331i.
What is the product of √(-56) and 3?
22.44993i
Calculate √(-56) = 7.48331i.
Then, multiply by 3: 7.48331i x 3 = 22.44993i.
Calculate (√(-56))^2.
-56
By definition, (√(-56))^2 = -56, as squaring the square root returns the original value.
What is the real part of √(-56)?
0
The square root of a negative number is purely imaginary, so the real part is 0.
Is √(-56) a real number?
No
The square root of a negative number is not a real number; it is an imaginary number.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.